Minimal dimension of faithful representations for p-groups

Abstract

For a group G, we denote by m faithful (G), the smallest dimension of a faithful complex representation of G. Let F be a non-Archimedean local field with the ring of integers O and the maximal ideal p. In this paper, we compute the precise value of m faithful (G) when G is the Heisenberg group over O/p n . We then use the Weil representation to compute the minimal dimension of faithful representations of the group of unitriangular matrices over O/p n and many of its subgroups. By a theorem of Karpenko and Merkurjev [7, Theorem 4.1], our result yields the precise value of the essential dimension of the latter finite groups.

On the minimal dimension of a faithful linear representation of a finite group

“…Repeating the same reasoning, we may also assume that χ 3 (1) ≤ |G| 1/2 /4 √ 2. Now, we bound χ 1 (1). With our usual notation and arguments, we may assume that |G : C| = 6 and one can see that…”

Kirillov's orbit method and polynomiality of the faithful dimension of $p$-groups

“…Similarly, U k (R) denotes the subgroup of unitriangular matrices in GL k (R), so that H 2k+1 (R) ⊆ U k+2 (R). In [3,Theorem 1.1] we proved that…”

“…Therefore we will only consider complex representations and use the shorthand m faithful (G) instead of m faithful,C (G). This work is a continuation of [3] in which the faithful dimension of a large class of p-groups was studied. Let us start by recalling some of the results from [3].…”

“…This work is a continuation of [3] in which the faithful dimension of a large class of p-groups was studied. Let us start by recalling some of the results from [3]. Let F be a non-Archimedean local field with a discrete valuation ν.…”

“…This problem is the central guiding principle of this work. Before stating our results, let us mention that the methods applied in [3] were based on a suitably adapted version of the Stone-von Neumann theory, whose application is mostly limited to groups of nilpotency class 2. In this paper, we will replace the Stone-von Neumann theory by Kirillov's orbit method for finite p-groups.…”